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Properties

Label 2-336-21.17-c3-0-17
Degree $2$
Conductor $336$
Sign $0.938 - 0.343i$
Analytic cond. $19.8246$
Root an. cond. $4.45248$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.18 − 0.273i)3-s + (2.97 + 5.16i)5-s + (17.8 − 5.06i)7-s + (26.8 + 2.83i)9-s + (−49.9 − 28.8i)11-s + 52.0i·13-s + (−14.0 − 27.5i)15-s + (17.3 − 30.1i)17-s + (26.7 − 15.4i)19-s + (−93.8 + 21.4i)21-s + (−21.8 + 12.6i)23-s + (44.7 − 77.5i)25-s + (−138. − 22.0i)27-s + 64.9i·29-s + (80.6 + 46.5i)31-s + ⋯
L(s)  = 1  + (−0.998 − 0.0526i)3-s + (0.266 + 0.461i)5-s + (0.961 − 0.273i)7-s + (0.994 + 0.105i)9-s + (−1.36 − 0.789i)11-s + 1.11i·13-s + (−0.241 − 0.474i)15-s + (0.247 − 0.429i)17-s + (0.322 − 0.186i)19-s + (−0.974 + 0.222i)21-s + (−0.198 + 0.114i)23-s + (0.357 − 0.620i)25-s + (−0.987 − 0.157i)27-s + 0.415i·29-s + (0.467 + 0.269i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.343i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.938 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.938 - 0.343i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.938 - 0.343i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.432489134\)
\(L(\frac12)\) \(\approx\) \(1.432489134\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (5.18 + 0.273i)T \)
7 \( 1 + (-17.8 + 5.06i)T \)
good5 \( 1 + (-2.97 - 5.16i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (49.9 + 28.8i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 52.0iT - 2.19e3T^{2} \)
17 \( 1 + (-17.3 + 30.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-26.7 + 15.4i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (21.8 - 12.6i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 64.9iT - 2.43e4T^{2} \)
31 \( 1 + (-80.6 - 46.5i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-162. - 280. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 282.T + 6.89e4T^{2} \)
43 \( 1 - 441.T + 7.95e4T^{2} \)
47 \( 1 + (-291. - 505. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (199. + 115. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-88.3 + 153. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-589. + 340. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (56.2 - 97.4i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 705. iT - 3.57e5T^{2} \)
73 \( 1 + (240. + 138. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-275. - 476. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.12e3T + 5.71e5T^{2} \)
89 \( 1 + (643. + 1.11e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 76.4iT - 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.08056181082462529930831530687, −10.58860339906118830161065113468, −9.532435887008001942183249634374, −8.147049610603373635825579362608, −7.28911338423555680986666240589, −6.23541725963734328253762565200, −5.25434370283499703923804783124, −4.36027133536162930789853609903, −2.55582354846121134319957495573, −0.960706977196883016008245072474, 0.800728411589149931308187614537, 2.28999626084814553709312478498, 4.29165566049828584724525260416, 5.33707293695985888624683629112, 5.72930391950318077007743540002, 7.41471489261301900774092295174, 8.006064986245498192071018497970, 9.375560204273580650345526701187, 10.42198566428703761697149323713, 10.90334569852383537640836899286

Graph of the $Z$-function along the critical line